Duke Mathematical Journal

Multidimensional boundary layers for a singularly perturbed Neumann problem

Andrea Malchiodi and Marcelo Montenegro

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Abstract

We continue the study of [34], proving concentration phenomena for the equation − ε2 Δu + u = up in a smooth bounded domain Ω ⊆ $\mathbb{R}^n and with Neumann boundary conditions. The exponent p is greater than or equal to 1, and the parameter ε is converging to zero. For a suitable sequence εj → 0, we prove the existence of positive solutions uj concentrating at the whole boundary of Ω or at some of its components.

Article information

Source
Duke Math. J. Volume 124, Number 1 (2004), 105-143.

Dates
First available: 30 July 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1091217476

Digital Object Identifier
doi:10.1215/S0012-7094-04-12414-5

Mathematical Reviews number (MathSciNet)
MR2072213

Zentralblatt MATH identifier
02103759

Subjects
Primary: 35B25: Singular perturbations 35B34: Resonances 35J20: Variational methods for second-order elliptic equations 35J60: Nonlinear elliptic equations

Citation

Malchiodi, Andrea; Montenegro, Marcelo. Multidimensional boundary layers for a singularly perturbed Neumann problem. Duke Mathematical Journal 124 (2004), no. 1, 105--143. doi:10.1215/S0012-7094-04-12414-5. http://projecteuclid.org/euclid.dmj/1091217476.


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References

  • S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions, I, Comm. Pure Appl. Math. 12 (1959), 623--727.
  • A. Ambrosetti and M. Badiale, Homoclinics: Poincaré-Melnikov type results via a variational approach, Ann. Inst. H. Poincaré Anal. Non Linéaire 15 (1998), 233--252.
  • A. Ambrosetti, M. Badiale, and S. Cingolani, Semiclassical states of nonlinear Schrödinger equations, Arch. Rational Mech. Anal. 140 (1997), 285--300.
  • A. Ambrosetti, A. Malchiodi, and W.-M. Ni, Solutions, concentrating on spheres, to symmetric singularly perturbed problems, C. R. Math. Acad. Sci. Paris 335 (2002), 145--150.
  • --. --. --. --., Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, I, Comm. Math. Phys. 235 (2003), 427--466.
  • --------, Singularly perturbed elliptic equations with symmetry: Existence of solutions concentrating on spheres, II, to appear in Indiana Univ. Math. J.
  • A. Ambrosetti, A. Malchiodi, and S. Secchi, Multiplicity results for some nonlinear Schrödinger equations with potentials, Arch. Ration. Mech. Anal. 159 (2001), 253--271.
  • M. Badiale and T. D'Aprile, Concentration around a sphere for a singularly perturbed Schrödinger equation, Nonlinear Anal. Ser. A: Theory Methods 49 (2002), 947--985.
  • M. Ben Ayed, K. El Mehdi, and M. Hammami, A nonexistence result for Yamabe type problems on thin annuli, Ann. Inst. H. Poincaré Anal. Non Linéaire 19 (2002), 715--744.
  • V. Benci and T. D'Aprile, The semiclassical limit of the nonlinear Schrödinger equation in a radial potential, J. Differential Equations 184 (2002), 109--138.
  • S. Brendle, On solutions to the Ginzburg-Landau equations in higher dimensions, preprint.
  • --------, On the construction of solutions to the Yang-Mills equations in higher dimensions, preprint.
  • I. Chavel, Riemannian Geometry---A Modern Introduction, Cambridge Tracts in Math. 108, Cambridge Univ. Press, Cambridge, 1993.
  • S. Cingolani and A. Pistoia, Nonexistence of single blow-up solutions for a nonlinear Schrödinger equation involving critical Sobolev exponent, Z. Angew. Math. Phys. 55 (2004), 201--215.
  • E. N. Dancer, New solutions of equations on $\mathbbR^n$, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 30 (2001), 535--563.
  • --------, Stable and finite Morse index solutions on $\mathbbR^n$ or on bounded domains with small diffusion, to appear in Trans. Amer. Math. Soc.
  • E. N. Dancer and S. Yan, Multipeak solutions for a singularly perturbed Neumann problem, Pacific J. Math. 189 (1999), 241--262.
  • T. D'Aprile, On a class of solutions with non-vanishing angular momentum for nonlinear Schrödinger equations, Differential Integral Equations 16 (2003), 349--384.
  • M. Del Pino, P. Felmer, and M. Kowalczyk, Boundary spikes in the Gierer-Meinhardt system, Commun. Pure Appl. Anal. 1 (2002), 437--456.
  • M. Del Pino, P. Felmer, and J. Wei, On the role of mean curvature in some singularly perturbed Neumann problems, SIAM J. Math. Anal. 31 (1999), 63--79.
  • A. Floer and A. Weinstein, Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential, J. Funct. Anal. 69 (1986), 397--408.
  • A. Gierer and H. Meinhardt, A theory of biological pattern formation, Kybernetik (Berlin) 12 (1972), 30--39.
  • D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Grundlehren Math. Wiss. 224, Springer, Berlin, 1977.
  • M. Grossi, A. Pistoia, and J. Wei, Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory, Calc. Var. Partial Differential Equations 11 (2000), 143--175.
  • C. Gui, Multipeak solutions for a semilinear Neumann problem, Duke Math. J. 84 (1996), 739--769.
  • C. Gui and J. Wei, On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems, Canad. J. Math. 52 (2000), 522--538.
  • C. Gui, J. Wei, and M. Winter, Multiple boundary peak solutions for some singularly perturbed Neumann problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 17 (2000), 47--82.
  • D. Iron, M. J. Ward, and J. Wei, The stability of spike solutions to the one-dimensional Gierer-Meinhardt model, Phys. D 150 (2001), 25--62.
  • T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Grundlehren Math. Wiss. 132, Springer, Berlin, 1976.
  • Y. Y. Li, On a singularly perturbed equation with Neumann boundary condition, Comm. Partial Differential Equations 23 (1998), 487--545.
  • Y. Li and L. Nirenberg, The Dirichlet problem for singularly perturbed elliptic equations, Comm. Pure Appl. Math. 51 (1998), 1445--1490.
  • C.-S. Lin, W.-M. Ni, and I. Takagi, Large amplitude stationary solutions to a chemotaxis system, J. Differential Equations 72 (1988), 1--27.
  • A. Malchiodi, Adiabatic limits for some Newtonian systems in $\mathbbR^n$, Asymptot. Anal. 25 (2001), 149--181.
  • A. Malchiodi and M. Montenegro, Boundary concentration phenomena for a singularly perturbed elliptic problem, Comm. Pure Appl. Math. 55 (2002), 1507--1568.
  • W.-M. Ni, Diffusion, cross-diffusion, and their spike-layer steady states, Notices Amer. Math. Soc. 45 (1998), 9--18.
  • W.-M. Ni and I. Takagi, On the shape of least-energy solutions to a semilinear Neumann problem, Comm. Pure Appl. Math. 44 (1991), 819--851.
  • --. --. --. --., Locating the peaks of least-energy solutions to a semilinear Neumann problem, Duke Math. J. 70 (1993), 247--281.
  • W.-M. Ni, I. Takagi, and E. Yanagida, ``Stability of least energy patterns of the shadow system for an activator-inhibitor model'' in Recent Topics in Mathematics Moving Toward Science and Engineering, Japan J. Indust. Appl. Math. 18 (2001), 259--272.
  • Y. G. Oh, On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential, Comm. Math. Phys. 131 (1990), 223--253.
  • F. Pacard and M. Ritoré, From constant mean curvature hypersurfaces to the gradient theory of phase transitions, preprint.
  • J. Shatah and C. Zeng, Periodic solutions for Hamiltonian systems under strong constraining forces, J. Differential Equations 186 (2002), 572--585.
  • J. Shi, Semilinear Neumann boundary value problems on a rectangle, Trans. Amer. Math. Soc. 354 (2002), 3117--3154.
  • A. M. Turing, The chemical basis of morphogenesis, Phil. Trans. Roy. Soc. London Ser. B Bio. Sci. 237 (1952), 37--72.
  • Z. Q. Wang, On the existence of multiple, single-peaked solutions for a semilinear Neumann problem, Arch. Rational Mech. Anal. 120 (1992), 375--399.
  • J. Wei, On the boundary spike layer solutions to a singularly perturbed Neumann problem, J. Differential Equations 134 (1997), 104--133.